1. Field of the Invention
This invention relates to the field of tiles and tilings. The field includes the familiar floor and kitchen-counter top tiles and tilings of commerce and their like, but also extends to the sometimes more abstract areas of art, design, and mathematics.
Some Definitions
I have adapted a few of the notions and definitions of the mathematics of tiles and tilings as follows. A tile is a two-dimensional closed shape which fits together edge-to-edge with other different or similar two-dimensional shapes, as do jig-saw puzzle pieces or bricks, to cover a flat surface of indefinite extent. Such a covering is called a tiling if it has no gap between tiles nor any overlap of one tile on another. Adding thickness to a two dimensional tile will make it a three dimensional object which is also called a tile. A tile or a set of tiles is said to tile the plane if indefinitely large numbers of duplicates of the tile or of the members of the set of tiles can fit together without gap or lap in a tiling. The term the plane refers to the flat indefinitely extensive plane of Euclidian geometry. As a verb, tile means to form a tiling.
A figurative tile is one whose shape is the recognizable outline, or figure, of a person or an animal. A figurative tiling is a tiling composed of such figurative tiles.
Figurative and zoomorphic are very much the same. Zoomorphic includes anthropomorphic. A family of zoomorphic tiles comprises all and only tiles whose four edges are formed by a particular amphographic line and its mirror image.
A variably assemblable tiling is one whose tiles function so as to fit together or to interlock with one another in a variety of different ways, allowing a plurality of different tilings to be made. Perhaps the simplest tile to form such a plurality is the common brick with its many different arrangements and patternings in walls and pavings. Sets of curved sided tiles that are variably assemblable are somewhat more difficult to design, as may be seen in U.S. Pat. No. 4,217,740 of Aug. 19, 1980 to Assanti.
An amphographic line is a line each side of which draws a different part of the outline of the same figure, and does so between the vertices of an ancestral straight-line geometric figure of a sort chosen so that the completed figurative outline will, when replicated, tile the plane. It can be said, then, that an amphographic line such as is used in single-figure periodic tilings is a line that differs from ordinary lines in doing double the duty of the usual outline. The invention described herein allows an effective redoubling of the number of things drawn by certain carefully devised amphographic lines so that they become lines that do quadruple duty. Each side of such a line depicts different parts of two different figures.
An ancestral rhombus can be thought of as an underlying, invisible, geometric determiner of vertex locations. Dotted lines are used herein to show ancestral rhombuses.
Expressions such as 60/120 or 45/135 describe rhombuses having internal or vertex angles as indicated, eg.: 60 degrees and 120 degrees, or 45 degrees and 135 degrees.
By zoomorphic outline of a different sort is meant not merely a fatter or a skinnier version, (based on a fatter or skinnier ancestral rhombus), of a given zoomorphic outline, but rather an outline of a different zoological character, in the way that a bat is different from a lizard, or, (as seen in the Mother and Baby Sea Turtles shown in FIGS. 10 and 12), in the way a juvenile's shape is different from that of an adult.
2. Prior Art
U.S. Pat. No. 4,133,152 of Jan. 9, 1979 to Penrose shows a figurative and variably assemblable set of two tiles that has since become known as "Penrose's chickens". This is the only known variably assemblable figurative tile set that is not the work of the present inventor, John A. L. Osborn.